Saturday, January 11, 2014

Solving Hilbert’s sixth problem

The fundamental relationship between ontology and dynamic

Last time we have derived the Leibnitz identity which is the
root cause for information conservation in nature (from this one can derive
unitarity in quantum mechanics for example). How does Leibnitz identity hold
under tensor composition?

The quantum reconstruction argument is categorical, meaning
it is naturally expressed in terms of category theory but whenever possible we will stress a more physical point of view. So now let
us introduce a composability category U(⊗,R, ρ,…) where ⊗ is the tensor product
which combines two physical systems A and B into a larger system: A⊗B. As an example, A⊗B
could be a hydrogen atom, where A is the electron, and B is the proton.

R represents the
real number field and we pick R over
any other mathematical field because we want to be able to compute probabilities in the usual way. ρ,… belong to our unspecified set of
local operations {o}. This composability category has as the identity element
the chosen field R (U⊗R = R⊗U = U)and elements of R will be understood
as arbitrary constant functions.

Now we can
see how ρ
acts on a constant function 1. Using the Leibnitz
identity:

f ρ 1 = f ρ (1 x 1) = (f ρ 1) x
1 + 1 x (f ρ 1) = 2x (f ρ 1)

and so we have in general that f ρ1 = 0 for
any function f. (This is very natural, we just stated in a fancy abstract way
that the derivation of a constant function is zero)

Using the tensor product and using the unit of the
composability category we have:

(f ⊗1) ρ12 (g⊗1) = (f ρ g) ⊗1 = (f ρ g)

where ρ12 is the bipartite product
ρ.

The bipartite (and in general the n-partite) products must
be build out of the available products in {o}. If in our universe of discourse the collection {o} of the available
products contains only the product ρ we have:

.(f ⊗1) ρ12 (g⊗1) = (f ρ g) ⊗(1 ρ 1) = 0 because (1 ρ 1) = 0

Hence the ρ product is trivial
if it exists by itself. There must exists at least another product θ to
have an interesting domain. If ρ corresponds to the dynamic θ
corresponds to ontology (observables).

Suppose that {o} contains only ρ and
θ. The bipartite products ρ12and θ12must be constructed out of ρ and θ. The most general way for
this is as follows:

Now the
goal is to determine the values of the parameters a,b,c,d,x,y,z,w. Since the
relationship is general, we can pick f1 =f2 = 1 and we can use the identity: 1 rho
g = 0;

Then in
the first relationship only c and d terms survive. If we normalize Theta
such that (1 theta 1) = 1 this demands c=1 and d=0. Similarly z=0, w = 1. Using
the same trick with g1 = g 2 = 1
demands b=1 y=0

If Theta
represents the algebra of observables and Alice and Bob form a bipartite system
(EPR pair), x=0 means that the observables
are separable. x != 0 means that the observables are affected by the dynamics
and the system can be entangled!!!

We can normalize
x to be +1,0,-1, but if we do preserve the dimensions, when it is not zero, x =
+/- ħ2/4.

Invariance
of the laws of nature under composability demands that x remains the same or,
equivalently that the Plank constant is the same for all quantum systems!

Now for
some references.

The core
ideas were developed by Emile Grgin and Aage Petersen at Yeshiva University in
the 70s (Aage Petersen was Bohr’s assistant and Bohr had the hunch that
classical and quantum mechanics share core features beyond the correspondence
principle)

Expanding
on Grgin’s original idea I wrote: http://arxiv.org/abs/1303.3935
which was uploaded on the archive only 11 days before a similar result by Anton
Kapustin of Caltech: http://arxiv.org/abs/1303.6917
Kapustin’s paper had the same old Grgin paper for inspiration and is written in the category
theory formulation. We worked independently and the papers are about 80%
identical in content notwithstanding that they look very different. Eliminating
(correctly and conclusively) the hyperbolic composability class was done in http://arxiv.org/abs/1311.6461 and
his lead to a major generalization of functional analysis (I’ll cover this in
subsequent posts). There are still more unpublished results.